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Atwell, J. A.; King, B. B.

Proper orthogonal decomposition for reduced basis feedback controllers for parabolic equations. (English) Zbl 0964.93032

Math. Comput. Modelling 33, No. 1-3, 1-19 (2001).
The authors consider the infinite horizon LQR (linear-quadratic regulator) problem and the LQG (linear-quadratic Gaussian) problem for the heat equation in space dimension 1 with 1-dimensional control, \[ {\partial y(t, x) \over \partial t}=\kappa {\partial^2 y(t, x) \over \partial x^2} + b(x)u(t), \quad u(t, 0)=u(t, X)=0 . \] In the LQR problem one minimizes a suitable quadratic functional of the state and the control over controls in \(L^2(0, \infty);\) the optimal control \(\bar u(t)\) is given by a feedback law involving the solution to an operator algebraic Riccati equation. The LQG problem only requires a state estimate instead of the full state, the feedback law similar to that for the LQR problem. The authors use finite element discretizations and apply the proper orthogonal decomposition method to two examples of input collections. In the first example, the proper orthogonal decomposition is directly applied to the finite element basis of linear \(B\)-splines; in the other, time snapshots are included. The objective is the design of low-order controllers.
Reviewer: Hector O.Fattorini (Los Angeles)

Cited in 63 Documents

MSC:

93B40 Computational methods in systems theory (MSC2010)
93C20 Control/observation systems governed by partial differential equations
93B52 Feedback control
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs

Keywords:

proper orthogonal decomposition; principal component analysis; heat equation; feedback control; linear-quadratic regulator problem; finite element discretizations
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\textit{J. A. Atwell} and \textit{B. B. King}, Math. Comput. Modelling 33, No. 1--3, 1--19 (2001; Zbl 0964.93032)
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References:

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